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The problem in the image reads:

"The sides EF, FD, and DE of a triangle DEF are produced in order, forming three exterior angles ∠DFP, ∠FDR, and ∠FER respectively. Prove that ∠DFP + ∠FDR + ∠FER = 360°."

### Solution:

We are given a triangle \( DEF \) with exterior angles \( \angle DFP \), \( \angle FDR \), and \( \angle FER \). Exterior angles are the angles formed when the sides of the triangle are extended outward.

#### Step-by-Step Proof:

1. **Definition of Exterior Angles**:
   Each exterior angle is supplementary to the corresponding interior angle of the triangle. This means:
   \[
   \angle DFP + \angle EDF = 180^\circ
   \]
   \[
   \angle FDR + \angle FDE = 180^\circ
   \]
   \[
   \angle FER + \angle EFD = 180^\circ
   \]

2. **Sum of Interior Angles**:
   The sum of the interior angles of any triangle is always:
   \[
   \angle EDF + \angle FDE + \angle EFD = 180^\circ
   \]

3. **Adding the Exterior Angles**:
   Adding the exterior angles:
   \[
   (\angle DFP + \angle FDR + \angle FER)
   \]
   Using the fact that each exterior angle forms a linear pair with the corresponding interior angle, and the sum of interior angles of a triangle is \( 180^\circ \), we get:
   \[
   (\angle DFP + \angle EDF) + (\angle FDR + \angle FDE) + (\angle FER + \angle EFD) = 3 \times 180^\circ
   \]
   This simplifies to:
   \[
   \angle DFP + \angle FDR + \angle FER + 180^\circ = 360^\circ
   \]
   
   Since \( \angle EDF + \angle FDE + \angle EFD = 180^\circ \), the sum of the exterior angles is:
   \[
   \angle DFP + \angle FDR + \angle FER = 360^\circ
   \]

Thus, the sum of the exterior angles is proven to be \( 360^\circ \).

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Would you like more details on any step, or have further questions? 

Here are 5 related questions to expand on this:

1. What are the properties of exterior angles in polygons?
2. Can the exterior angles of any polygon add up to more than 360°?
3. How do exterior angles help in solving triangle-related problems?
4. How would this theorem apply if the triangle were not extended?
5. What is the relation between interior and exterior angles in other polygons?

**Tip:** Exterior angles always sum to 360°, regardless of the shape of the polygon.

The sides EF, FD, and DE of a triangle DEF are produced in order, forming three exterior angles ∠DFP, ∠FDR, and ∠FER respectively. Prove that ∠DFP + ∠FDR + ∠FER = 360°.

Understand how to prove that the sum of exterior angles ∠DFP, ∠FDR, and ∠FER of triangle DEF equals 360°. This problem explores the exterior angle theorem and the sum of interior angles in a triangle, ideal for students in Grades 8-10.

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